Circles A and B have the following equations #(x -4 )^2+(y +3 )^2= 25 # and #(x +4 )^2+(y -1 )^2= 16 #. What is the greatest possible distance between a point on circle A and another point on circle B?

Question

Circles A and B have the following equations #(x   -4  )^2+(y   +3   )^2=  25  # and #(x    +4   )^2+(y   -1   )^2=  16  #.  What is the greatest possible distance between a point on circle A and another point on circle B?

Allison Allen · Accepted Answer

#color(green)(9+4sqrt(5))#

The greatest possible distance will be along the extended line joining the centers of the two circles, between the farthest points of each circle from the other.

Circle A:
#(x-4)^2+(y+3)^2=25#
#color(white)("XXX")#has its center at #(4,-3)#
#color(white)("XXX")#and a radius of #5#

Circle B
#(x+4)^2+(y-1)^2=16#
#color(white)("XXX")#has its center at #(-4,1)#
#color(white)("XXX")#and a radius of #4#

The distance between the two centers is
#color(white)("XXX")sqrt((4-(-4))^2+(-3-1)^2)=sqrt(8^2+4^2)=4sqrt(5)#

The greatest distance between two points on these circles would be
#color(white)("XXX")5+4sqrt(5)+4 = 9+4sqrt(5)#

Aurora Ayala · Answer

undefined The greatest possible distance between a point on circle A and another point on circle B is the sum of their radii, which is $ \sqrt{25} + \sqrt{16} = 5 + 4 = 9 $ units.